A monetary value for initial information in portfolio optimization

Insider Information, Arbitrage and Optimal Portfolio and Consumption Policies

Many engineering decisions involve uncertainty and require tradeoffs between multiple attributes. Often it is possible to reduce or eliminate uncertainty through data collection, or analysis, or both. Although it generally leads to better outcomes, reducing uncertainty may increase cost. This metadecision of whether to reduce the uncertainty at a given cost is undertaken in the form of Value of Information (VOI) studies. Previous work has considered value of information in singleattribute cases, however, studies of the more challenging multiattribute case are less common. Multiattribute problems can have numerous uncertain variables across alternatives, not all of them easily measurable in money amounts, which makes the VOI analysis more complicated. This paper presents a simulation-based approach for calculating and interpreting the VOI in generic multiattribute decision problems and describes a simulation study that we conducted to explore how the decision problem parameters affect the VOI. The relationship between the VOI and the gain in expected utility (utility-gain) that will result from new information is also discussed. This relationship is not simple and yields some expected as well as counterintuitive results. For example, our results suggest that non-linearity of the utility functions may lead to different VoI calculation for the same utility-gain, depending on the relative position of the alternatives. Finally, this paper discusses the sensitivity of the utility from the decision as a function of decision problem parameters and shows that the magnitude of change in VoI can be predicted from these sensitivities. A vehicle alignment problem in an automotive assembly line is presented to demonstrate the approach. The results show that VoI calculation is a challenging problem that is very much contingent on the parameters of the problem. Our results provide insights into the benefits of different ways to evaluate the value of information in multiattribute problems.

The paper studies connections between risk aversion and martingale measures in a discrete‐time incomplete financial market. An investor is considered whose attitude toward risk is specified in terms of the index b of constant proportional risk aversion. Then dynamic portfolios are admissible if the terminal wealth is positive. It is assumed that the return (risk) processes are bounded. Sufficient (and nearly necessary) conditions are given for the existence of an optimal dynamic portfolio which chooses portfolios from the interior of the set of admissible portfolios. This property leads to an equivalent martingale measure defined through the optimal dynamic portfolio and the index 0 < b≤ 1. Moreover, the option pricing formula of Davis is given by this martingale measure. In the case of b= 1; that is, in the case of the log‐utility, the optimal dynamic portfolio defines the numéraire portfolio.

The paper studies connections between arbitrage and utility maximization in a discrete-time financial market. The market is incomplete. Thus one has several choices of equivalent martingale measures to price contingent claims. Davis determines a unique price for a contingent claim which is based on an optimal dynamic portfolio by use of a `marginal rate of substitution' argument. Here conditions will be given such that this price is determined by a martingale measure and thus by a consistent price system. The underlying utility function U is defined on the positive half-line. Then dynamic portfolios are admissible if the terminal wealth is positive. In case of the logarithmic utility function, the optimal dynamic portfolio is the numeraire portfolio.

The aim of this paper is to find optimal portfolio strategies for an n-stock extension of the stochastic volatility model proposed in Ref. [2]. It is a modification of [19], and characterizes the dependence by the use of a factor structure. The idea of a factor structure is that the diffusion components of the stocks contain one Brownian motion that is unique for each stock, and a few Brownian motions that all stocks share. Our model can obtain strong correlations between the returns for different stocks without affecting their marginal distributions. This was not possible in Ref. [19]. Further, the number of model parameters does not grow too fast as the number of stocks n grows. We use dynamic programming to solve Merton’s optimization problem for power utility, with utility drawn from terminal wealth. Optimal strategies for n stocks are obtained. We also discuss how to estimate the model from data. This is illustrated with an example.

In this article, we are interested in an investment problem with stochastic volatilities and Portfolio constraints on proportions. We model the risky assets by jump diffusion processes and we consider a power utility function. The objective is to maximize the expected utility from terminal wealth. The value function is known to be a viscosity solution of an integro-differential Hamilton-Jacobi-Bellman (HJB) equation. Thanks to an exponential transformation, we reduce the nonlinearity of the HJB equation to a semi-linear equation. We prove existence of a smooth solution to the latter equation and we state a verification theorem which relates this solution to the value function. We derive the backward stochastic differential equation (BSDE) associated with the semi-linear equation. We propose a numerical scheme for the resolution of the BSDE based on iterative regressions on function bases and Monte-Carlo method. We illustrate an example which shows the importance of this reduction for the optimal portfolio numerical study. Therefore, we compute the optimal investment strategy by solving the associated BSDE.

We consider an investor facing a classical portfolio problem of optimal investment in a log-Brownian stock and a fixed-interest bond, but constrained to choose portfolio and consumption strategies that reduce a dynamic shortfall risk measure. For continuous- and discrete-time financial markets we investigate the loss in expected utility of intermediate consumption and terminal wealth caused by imposing a dynamic risk constraint. We derive the dynamic programming equations for the resulting stochastic optimal control problems and solve them numerically. Our numerical results indicate that the loss of portfolio performance is not too large while the risk is notably reduced. We then investigate time discretization effects and find that the loss of portfolio performance resulting from imposing a risk constraint is typically bigger than the loss resulting from infrequent trading.

Portfolio Optimization Under Convex Incentive Schemes

Portfolio Optimization in a Defaultable LLvy Driven Market Model

Optimal portfolio and insurance strategy with biometric risks, habit formation and smooth ambiguity

The study investigates the value of additional information on the response function to soil salinity of a given crop (potatoes), with regard to a stochastic long‐run optimization model for utilization of saline water in a single‐farm framework. The analysis provides a conceptual and methodological framework for investigating the expected value of sample information (evsi), as well as an efficient tool for empirical application. Although a few approximations have been used, the results provide an estimate of evsi and indicate the need for additional information.

Optimal decision on dynamic insurance price and investment portfolio of an insurer

Mean–variance asset–liability management with asset correlation risk and insurance liabilities

O presente trabalho investigou os efeitos de quantias de dinheiro descritas como a alternativa de risco e da instrução sobre o valor subjetivo do reforço. Oito estudantes realizaram escolhas hipotéticas entre quantias de dinheiro certas ou imediatas que eram ajustadas de acordo com o comportamento dos sujeitos, e cinco quantias de magnitudes maiores (V = R$10,00 a R$100.000,00) atrasadas (uma semana a 50 anos) ou probabilísticas (0,05 a 0,95). A instrução positiva descrevia as chances ou o atraso para receber cada quantia, enquanto a instrução negativa descrevia as chances de perder ou o tempo de espera para não perder as quantias. O valor subjetivo diminuiu com o aumento de V provável, e aumentou com o aumento de V atrasado. Os resultados individuais foram melhor descritos por uma função potência do que pela hiperbólica. A variação da instrução não produziu diferenças sistemáticas. Os resultados são evidências para uma interpretação diferente do efeito da magnitude do reforço sobre o desconto de quantias prováveis ou atrasadas.

The cost of obtaining good information regarding the various probability distributions needed for the solution of most stochastic decision problems is considerable. It is important to consider questions such as: (1) what minimal amounts of information are sufficient to determine optimal decision rules; (2) what is the value of obtaining knowledge of the actual realization of the random vectors; and (3) what is the value of obtaining some partial information regarding the actual realization of the random vectors. This paper is primarily concerned with questions two and three when the decision maker has an a priori knowledge of the joint distribution function of the random variables. Some remarks are made regarding results along the lines of question one. Mention is made of assumptions sufficient so that knowledge of means, or of means, variances, co-variances and n-moments are sufficient for the calculation of optimal decision rules. The analysis of the second question leads to the development of bounds on the value of perfect information. For multiperiod problems it is important to consider when the perfect information is available. Jensen's inequality is the key tool of the analysis. The calculation of the bounds requires the solution of nonlinear programs and the numerical evaluation of certain functions. Generally speaking, tighter bounds may be obtained only at the expense of additional information and computational complexity. Hence, one may wish to compute some simple bounds to decide upon the advisability of obtaining more information. For the analysis of the value of partial information it is convenient to introduce the notion of a signal. Each signal represents the receipt of certain information, and these signals are drawn from a given probability distribution. When a signal is received, it alters the decision maker's perception of the probability distributions inherent in his decision problem. The choice between different information structures must then take into account these probability distributions as well as the decision maker's preference function. A hierarchy of bounds may be determined for partial information evaluation utilizing the tools of the multiperiod perfect information case. However, the calculation of these bounds is generally considerably more dicult than the calculation of similar boulids in the perfect information case. Most of the analysis is directed towards problems in which the decision maker has a linear utility function over profits, costs or some other numerical variable. However, some of the bounds generalize to the case when the utility function is strictly increasing and concave.